Econ 33220
Homework 3
Please correct any obvious typos. Explain your reasoning.
Problem 1. Consider the VAR
where
1. Calculate the characteristic roots. Is the VAR cointegrated?
2. Write B as
where you should normalize V so that the top row is a a row of 1’s.
3. Calculate the Blanchard-Quah decomposition. Plot the impulse responses to the transitory and the permanent shock up to a horizon of k = 20.
4. Calculate the Cholesky decomposition with pencil and paper. To do so, state
and calculate A ∗ A'. Proceed to calculate A11, A12 and A22 in that se-quence. Plot the impulse responses to the two shocks up to a horizon of k = 20.
5. Parameterize all potential decompositions of Σ with S(θ)S(θ)' = Σ with θ ∈ [0, π) and
where
is the Cholesky decomposition of Σ. Explicitly state the equations for the four entries of S(θ). Argue why this parameterization is reasonable. Put differently, which decompositions are missing, and how can they be ob-tained from these S(θ)?
6. Suppose, the first time series are quantities and the second time series are prices in some market. We seek to identify demand and supply shocks us-ing sign restrictions. Draw a standard demand-supply diagram. Supposing that a supply shock shifts the supply curve and that a demand shock shifts the demand curve, argue, why a demand shock moves prices and quantities in the same direction, but a supply shock moves prices and quantities in opposite directions.
7. Based on that insight, find all θ ∈ [0, π), so that S11(θ) ≥ 0 and S12(θ) ≤ 0. Plot S21(θ) and S22(θ). Numerically find the interval θ ∈ [, ] so that S(θ) satisfies all four sign restrictions S11(θ) ≥ 0, S21(θ) ≥ 0 (the demand shock) and S12(θ) ≤ 0, S22(θ) ≥ 0 (the supply shock). Plot the impulse responses to the supply shock and the demand shock, for θ = and θ = ( both θ’s in one diagram) up to a horizon of k = 20. Where would the impulse responses for the other θ ∈ [, ] be?
Problem 2. This is a coding exercise. I recommend doing this in MATLAB, but feel free using some other software. From FRED and per the clickable links, download the max data range of CPI inflation πt per “edit graph”, “Units: per-cent change from year ago”, “Modify frequency: quarterly”, the log of real GDP xt per “edit graph”, “Formula: a”, “Units: Natural Log”, the Federal Funds Rate it per “edit graph”, “quarterly”, and the log of the NASDAQ Composite Index zt per “edit graph”, “Modify frequency: quarterly”, “Formula: a”, “Units: Natural Log”. Load these four series into, say, MATLAB and chop them so that they are all have the same data range, making that range as large as possible (i.e.,find the highest starting date and the lowest ending date, and chop all series to that). Let us enumerate these dates per t = 0, . . . , T.
We now have the data vector
We seek to estimate and for the VAR
(1)
where c ∈ R4 is a vector of constants, using OLSE (“ordinary least squares estimation”). Do this as follows. Consider the first row of (1), written explicitly as
πt = c1 + B1,1πt−1 + B1,2xt−1 + B1,3it−1 + B1,4zt−1 + ut,π (2)
where we seek to estimate β1 = [c1, B1,1, B1,2, B1,3, B1,4]'. Define the T × 1 column vector
the T × 1 column vector
and the T × 5 matrix
Argue that one can then write (2) as
π = Yβ1 + uπ
The estimate for β1 then is the usual OLSE formula
giving us the first row of . The estimate for the residuals is
Do this for all four rows. You then get a T × 4 matrix of estimated residuals
which you can then use to estimate
Find the Cholesky decomposition and use the third column a of A as the column representing a monetary policy shock. Calculate the impulse responses to a, using your estimated , for a horizon of 12 quarters. Describe what you see. Does it look like a monetary policy shock?